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Rigidity Theorems in Riemannian Geometry

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Geometric Methods in Inverse Problems and PDE Control

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 137))

Abstract

The purpose of this chapter is to survey some recent results and state open questions concerning the rigidity of Riemannian manifolds. The starting point will be the boundary rigidity and conjugacy rigidity problems. These problems are connected to many other problems (Mostow-Margulis type rigidity, isopectral problems, isoperimetric inequalities etc.). We will restrict our attention to those results that have a direct connection to the boundary rigidity problem (see Section 2) or the conjugacy rigidity problem (see Section 4). Even with that restriction the connections are numerous and the author was forced to select the topics covered here in accordance with rather subjective criteria. A few of the topics not covered are mentioned in Section 11 but there are others.

Supported by NSF grant DMS 99-71749 and CRDF grant.

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Authors and Affiliations

  1. Department of Mathematics, University of Pennsylvania, 19104, Philadelphia, PA, USA

    Christopher B. Croke

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  1. Christopher B. Croke
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Editors and Affiliations

  1. Department of Mathematics, University of Pennsylvania, 19104-6395, Philadelphia, PA, USA

    Christopher B. Croke

  2. Department of Mathematics, Rutgers University, 08903, New Brunswick, NJ, USA

    Michael S. Vogelius

  3. Department of Mathematics, University of Washington, 98195-4350, Seattle, WA, USA

    Gunther Uhlmann

  4. Department of Mathematics, University of Virginia, 22903, Charlottesville, VA, USA

    Irena Lasiecka

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Croke, C.B. (2004). Rigidity Theorems in Riemannian Geometry. In: Croke, C.B., Vogelius, M.S., Uhlmann, G., Lasiecka, I. (eds) Geometric Methods in Inverse Problems and PDE Control. The IMA Volumes in Mathematics and its Applications, vol 137. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-9375-7_4

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  • DOI: https://doi.org/10.1007/978-1-4684-9375-7_4

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